How To Determine Sample Size, Determining Sample Size

	Bookmark This Page Email This Page Format for Printing Cite This Article Submit an Article Read More Articles
	Related Tools & Articles Learn how to randomly sample your population to ensure no bias Sample size needed to represent the population Process versus sampling variation

Download Products Project Selection Process Pack Lean Six Sigma Champion Training Design For Six Sigma (DFSS) E-Book or Print

In order to prove that a process has been improved, you must measure the process capability before and after improvements are implemented. This allows you to quantify the process improvement (e.g. defect reduction or productivity increase) and translate the effects into an estimated financial result – something business leaders can understand and appreciate. If data is not readily available for the process, how many members of the population should be selected to ensure that the population is properly represented? If data has been collected, how do you determine if you have enough data?

Determining sample size is a very important issue because samples that are too large may waste time, resources and money, while samples that are too small may lead to inaccurate results. In many cases, we can easily determine the minimum sample size needed to estimate a process parameter, such as the population mean .

When sample data is collected and the sample mean is calculated, that sample mean is typically different from the population mean . This difference between the sample and population means can be thought of as an error. The margin of error is the maximum difference between the observed sample mean and the true value of the population mean :

where:

is known as the critical value, the positive value that is at the vertical boundary for the area of in the right tail of the standard normal distribution.

is the population standard deviation.

is the sample size.

Rearranging this formula, we can solve for the sample size necessary to produce results accurate to a specified confidence and margin of error.

This formula can be used when you know and want to determine the sample size necessary to establish, with a confidence of , the mean value to within . You can still use this formula if you don’t know your population standard deviation and you have a small sample size. Although it’s unlikely that you know when the population mean is not known, you may be able to determine from a similar process or from a pilot test/simulation.

Let’s put all this statistical mumbo-jumbo to work. Take for example that we would like to start an Internet Service Provider (ISP) and need to estimate the average Internet usage of households in one week for our business plan and model.

Sample Size Calculation Example

Problem
We would like to start an Internet Service Provider (ISP) and need to estimate the average Internet usage of households in one week for our business plan and model. How many households must we randomly select to be 95% sure that the sample mean is within 1 minute of the population mean . Assume that a previous survey of household usage has shown = 6.95 minutes.

Solution
We are solving for the sample size .

A 95% degree confidence corresponds to = 0.05. Each of the shaded tails in the following figure has an area of = 0.025. The region to the left of and to the right of = 0 is 0.5 – 0.025, or 0.475. In the Table of the Standard Normal () Distribution, an area of 0.475 corresponds to a value of 1.96. The critical value is therefore = 1.96.

The margin of error = 1 and the standard deviation = 6.95. Using the formula for sample size, we can calculate :

So we will need to sample at least 186 (rounded up) randomly selected households. With this sample we will be 95% confident that the sample mean will be within 1 minute of the true population of Internet usage.

Small (<30) Sample Size Calculation

The formula for the sample size necessary to produce results accurate to a specified confidence and margin of error is:

where:

is known as the critical value, the positive value that is at the vertical boundary for the area of in the right tail of the standard normal distribution.

is the population standard deviation.

is the sample size.

This formula can be used when you know and want to determine the sample size necessary to establish, with a confidence of , the mean value to within .

As a general rule of thumb, if your sample size is greater than 30, you can replace by the sample standard deviation s.

If your sample size is less than or equal to 30, the population must be normally distributed and you must know the population standard deviation in order to use the formula above. An alternate solution method is to use the Student t Distribution developed by William Gosset.

Rate This Article:  Current Rating: 3.90   Poor    Excellent                1    2    3     4    5   View all top-rated articles

Copyright � 2000-2010 iSixSigma – All Rights Reserved
Reproduction Without Permission Is Strictly Prohibited – Copyright Requests

---

Publish an Article: Do you have a Six Sigma tip, learning or case study?
Share it with the largest community of Six Sigma professionals, and be recognized by your peers.
It's a great way to promote your expertise and/or build your resume. Read more about submitting an article.

"The Bottom Line" Links

BEST SELLING PRODUCTS (iSixSigma Publications) Six Sigma Black Belt (DMAIC) Training Slides - 2009 Version! The 2009 Six Sigma Black Belt course includes over 40 more slides than the 2008 version. Contents include: 1,220 PowerPo... Certified Lean Six Sigma Green Belt Assessment Exam This assessment exam is useful for students interested in assessing their knowledge of Lean Six Sigma on the Green Belt ... Certified Lean Six Sigma Black Belt Assessment Exam Interested in assessing your knowledge of Lean Six Sigma? Preparing for certifications? Testing your students and traine... Six Sigma Green Belt Training Slides - 2009 Version The 2009 Six Sigma Green Belt course is comprised of: 1122 slides (over 70 more slides than the 2008 version)Instructor... Certified Lean Six Sigma Black Belt E-book In 670 pages learn everything within the Lean Six Sigma DMAIC body of knowledge to successfully achieve Black Belt certi... Six Sigma DMAIC Training Slides - 2009 Version The complete 2009 Lean Six Sigma DMAIC course prepares participants to perform the role of a LSS Black Belt; covering wh... Kaizen Workshop E-book This 150+ page ebook teaches key tools and techniques of Kaizen, as well as real application to enhance learning. Kaizen... Sell Your Products